Circle Theorems and Arc Relationships

Key Vocabulary

• Radius: Center to circle ($r$)
• Diameter: Through center, ear to ear ($d = 2r$)
• Chord: Segment with endpoints on the circle
• Secant: Line through two points on the circle
• Tangent: Line touching the circle at exactly one point
• Arc: Part of the circle's circumference

Central Angles and Arcs

A central angle has its vertex at the center.

$\text{Central angle} = \text{Intercepted arc}$

Arc length: $L = \frac{\theta}{360°} \cdot 2\pi r$

Sector area: $A = \frac{\theta}{360°} \cdot \pi r^2$

Inscribed Angles

An inscribed angle has its vertex ON the circle.

$\text{Inscribed angle} = \frac{1}{2} \times \text{Intercepted arc}$

Key Theorems:

1. Inscribed angles intercepting the same arc are congruent
2. An inscribed angle in a semicircle is $90°$
3. Opposite angles of an inscribed quadrilateral sum to $180°$

Tangent Theorems

1. A tangent is perpendicular to the radius at the point of tangency
2. Two tangent segments from the same external point are congruent

Chord Theorems

1. If two chords are equal, they are equidistant from the center
2. A radius perpendicular to a chord bisects the chord

Angle Relationships

| Vertex Location | Formula | |----------------|---------| | Center | $\angle = \text{arc}$ | | On circle | $\angle = \frac{1}{2}(\text{arc})$ | | Inside circle | $\angle = \frac{1}{2}(\text{arc}_1 + \text{arc}_2)$ | | Outside circle | $\angle = \frac{1}{2}(\text{arc}_1 - \text{arc}_2)$ |

Equation of a Circle

Standard form: $(x - h)^2 + (y - k)^2 = r^2$

Center: $(h, k)$, Radius: $r$

Example: $(x - 3)^2 + (y + 2)^2 = 25$ → Center $(3, -2)$, $r = 5$

Don't forget: Convert general form $x^2 + y^2 + Dx + Ey + F = 0$ to standard form by completing the square!